Synthetic Regression Data

Dive into Deep Learning · §2.3

Build a dataset with known generating parameters
to isolate implementation and optimization errors.

Why use synthetic data?

Motivation

On real data, a poor result may reflect model misspecification, an optimization or implementation error, or properties of the data.

Synthetic data specifies the generative law, so we can test whether a compatible method recovers known parameters:

\mathbf{y} = \mathbf{X}\mathbf{w}^* + b^* + \boldsymbol{\epsilon}, \qquad \boldsymbol{\epsilon}\sim\mathcal{N}(0,\sigma^2 I).

Agreement with \mathbf{w}^*,b^* supports the implementation on this controlled problem. Systematic disagreement indicates an optimization or implementation problem, provided the fitted model matches the generator.

The dataset lives in a DataModule (the object-oriented-design section): where the batches come from, kept separate from the model.

01

Generating the data

a DataModule that knows the ground truth

Generate data in a DataModule

Generating the data

Draw \mathbf{X}\sim\mathcal{N}(0,1), apply the true line, add tiny noise, all inside __init__ (n=2000 examples, two features):

class SyntheticRegressionData(d2l.DataModule):
    """Synthetic data for linear regression."""
    def __init__(self, w, b, noise=0.01, num_train=1000, num_val=1000, 
                 batch_size=32):
        super().__init__()
        self.save_hyperparameters()
        n = num_train + num_val
        self.X = d2l.randn(n, len(w))
        eps = d2l.randn(n, 1) * noise
        self.y = d2l.matmul(self.X, d2l.reshape(w, (-1, 1))) + b + eps

save_hyperparameters() stores every argument as an attribute.

Set and inspect the generating parameters

Generating the data

Instantiate with the true \mathbf{w}^*=[2,-3.4]^\top, b^*=4.2:

data = SyntheticRegressionData(w=d2l.tensor([2, -3.4]), b=4.2)

Each feature row is a vector in \mathbb{R}^2; each label is a scalar:

print('features:', data.X[0],'\nlabel:', data.y[0])

features: [0.49267077 0.50505114] 
label: [3.4740288]

The next two sections compare the fitted parameters with [2, -3.4] and 4.2, allowing for estimation error from the added noise.

02

Reading the data

minibatches, by hand and by framework

A minibatch sampler, by hand

Reading the data

The hand-written minibatch loader shuffles the indices (afresh on every training pass), then yield batch_size rows at a time (one batch is 32\times2 features, 32\times1 labels).

def get_dataloader(self, train):
    if train:
        indices = list(range(0, self.num_train))
        # The examples are read in random order
        random.shuffle(indices)
    else:
        indices = list(range(self.num_train, self.num_train+self.num_val))
    for i in range(0, len(indices), self.batch_size):
        batch_indices = d2l.tensor(indices[i: i+self.batch_size])
        yield self.X[batch_indices], self.y[batch_indices]

This implementation is transparent, but it keeps all data in memory, iterates in single-threaded Python, and does not prefetch batches.

03

The built-in loader

the same interface with framework data-loading features

Hand the work to the framework

The built-in loader

The framework loader can shuffle, prefetch, and parallelize loading. First, wrap the tensors:

@d2l.add_to_class(d2l.DataModule)
def get_tensorloader(self, tensors, train, indices=slice(0, None)):
    tensors = tuple(a[indices] for a in tensors)
    dataset = gluon.data.ArrayDataset(*tensors)
    return gluon.data.DataLoader(dataset, self.batch_size,
                                 shuffle=train)

Then make get_dataloader use it for the training or validation split:

@d2l.add_to_class(SyntheticRegressionData)
def get_dataloader(self, train):
    i = slice(0, self.num_train) if train else slice(self.num_train, None)
    return self.get_tensorloader((self.X, self.y), train, i)

Same interface, drop-in

The built-in loader

The caller sees an identical protocol, one minibatch at a time:

X, y = next(iter(data.train_dataloader()))
print('X shape:', X.shape, '\ny shape:', y.shape)

X shape: (32, 2) 
y shape: (32, 1)

And it knows its own length, so len(dl) is the batches per epoch (\lceil 1000/32\rceil = 32: 31 full, one of 8):

len(data.train_dataloader())

32

Recap

Wrap-up

• Synthetic data fixes the answer up front (\mathbf{w}^*=[2,-3.4], b^*=4.2), enabling a controlled check of a compatible training method.
• A DataModule packages where batches come from, reusable across models.

• Hand-rolled vs. built-in loader: one protocol; the framework version shuffles, prefetches, parallelizes.
• Watch the last batch: a loader either keeps the partial final minibatch or drops it (32 vs. 31 batches here).